Vol. 303, No. 2, 2019

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Polarization, sign sequences and isotropic vector systems

Gergely Ambrus and Sloan Nietert

Vol. 303 (2019), No. 2, 385–399
Abstract

We determine the order of magnitude of the n-th p-polarization constant of the unit sphere Sd1 for every n,d 1 and p > 0. For p = 2, we prove that extremizers are isotropic vector sets, whereas for p = 1, we show that the polarization problem is equivalent to that of maximizing the norm of signed vector sums. Finally, for d = 2, we discuss the optimality of equally spaced configurations on the unit circle.

Keywords
polarization problems, discrete potentials, Chebyshev constants, isotropic vectors sets, tight frames, vector sums
Mathematical Subject Classification 2010
Primary: 52A40
Secondary: 41A17
Milestones
Received: 23 April 2019
Revised: 19 June 2019
Accepted: 19 June 2019
Published: 4 January 2020
Authors
Gergely Ambrus
Alfréd Rényi Institute of Mathematics
Hungarian Academy of Sciences
Budapest
Hungary
Sloan Nietert
Department of Computer Science
Cornell University
Ithaca, NY
United States